How to Evaluate Integrals- Calculus Tutorial
What Integration Actually Is
Integration is the reverse of differentiation. That's it. If you know how to take derivatives, you already understand half of calculus. The other half is recognizing which technique to apply and not screwing up the algebra.
Most students struggle with integration not because the concepts are hard, but because they expect it to be straightforward. It isn't. Differentiation follows predictable rules. Integration requires pattern recognition and practice.
Basic Integration Rules You Must Know
Before attempting anything complex, these need to be automatic:
- ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1
- ∫ 1/x dx = ln|x| + C
- ∫ eˣ dx = eˣ + C
- ∫ sin(x) dx = -cos(x) + C
- ∫ cos(x) dx = sin(x) + C
The +C isn't optional. Forgetting the constant of integration will cost you points on every exam.
U-Substitution: Your First Line of Attack
U-substitution works when you spot a composite function. The chain rule differentiated those. Now you undo it.
When to Use It
Look for function-within-a-function patterns. If you see something like f(g(x)) · g'(x), u-substitution will handle it.
How to Do It
- Pick a piece of the integrand to call "u" — usually the inner function
- Take du/dx and solve for dx
- Substitute everything
- Integrate in terms of u
- Replace u with the original expression
Example: ∫ 2x·cos(x²) dx
Let u = x². Then du = 2x dx. The integral becomes ∫ cos(u) du = sin(u) + C = sin(x²) + C.
Check your answer by differentiating. If you don't get the original integrand, you made a mistake.
Integration by Parts: When Substitution Fails
Integration by parts handles products where u-substitution doesn't fit. The formula comes from the product rule:
∫ u dv = uv - ∫ v du
Choosing u and dv
Use LIATE as a guide:
- Logarithmic
- Inverse trigonometric
- Algebraic
- Trigonometric
- Exponential
Pick u based on whichever type appears first. Let dv be the remaining part.
Example: ∫ x·eˣ dx
u = x, dv = eˣ dx
du = dx, v = eˣ
∫ x·eˣ dx = x·eˣ - ∫ eˣ dx = x·eˣ - eˣ + C
Sometimes you need to apply integration by parts twice. Sometimes you end up with the original integral on both sides — solve for it algebraically.
Definite vs Indefinite Integrals
Indefinite integrals give you a family of functions plus a constant. No bounds.
Definite integrals give you a number. You evaluate at the upper bound, subtract the value at the lower bound, and the +C cancels out.
With definite integrals, you can also change variables — just remember to adjust your bounds. Many students forget this and get the wrong answer.
Common Integrals Reference Table
| Integral | Result |
|---|---|
| ∫ xⁿ dx | (xⁿ⁺¹)/(n+1) + C |
| ∫ 1/x dx | ln|x| + C |
| ∫ eˣ dx | eˣ + C |
| ∫ aˣ dx | aˣ/ln(a) + C |
| ∫ sin(x) dx | -cos(x) + C |
| ∫ cos(x) dx | sin(x) + C |
| ∫ sec²(x) dx | tan(x) + C |
| ∫ csc²(x) dx | -cot(x) + C |
| ∫ 1/(1+x²) dx | arctan(x) + C |
| ∫ 1/√(1-x²) dx | arcsin(x) + C |
Getting Started: A Practical Approach
- Know your basics cold. Memorize the derivative rules first. If you can differentiate anything instantly, integrating becomes pattern matching instead of guessing.
- Try substitution first. It's the most common technique and handles most problems you'll encounter early on.
- Check your work. Differentiate your answer. You should get the original integrand. If not, find the error.
- Don't memorize formulas — understand them. The integral of 1/x is ln|x| because the derivative of ln|x| is 1/x. That's the reasoning. Build from there.
- Practice with variety. Do problems from different sources. Exams mix techniques. If you only practice easy integrals, you'll freeze when things get messy.
Common Mistakes That Cost Points
- Forgetting +C on indefinite integrals
- Choosing the wrong u in integration by parts
- Not adjusting bounds after u-substitution in definite integrals
- Dropping absolute value bars with ln functions
- Algebra errors in simplifying after integration
- Assuming every integral has a closed-form solution — some don't
When to Move Beyond Basics
Partial fractions handle rational functions where the denominator doesn't factor nicely into linear terms. Trigonometric substitution handles radicals with squares and sums or differences. These come later and build on the fundamentals above.
Master the basics first. Most calculus courses test u-substitution and integration by parts heavily. If you can do those reliably, you'll pass the exam.